Throughout our probabilistic arguments we make often use of the following Chernoff inequalities (see e.g. [1]}) that help to verify that a given binomial random variable \(X \sim \operatorname{Bin}(n,p)\) , where each of \(n\)  independent rounds has probability \(p\)  of being successful, is typically close to its expectation \(\mathbb {E}(X)=np\) .
